This notebook uses a number of relatively simple quantitative metrics to operationalize the integrative complexity of numeral (i.e., number word) systems across languages, which can be taken as a measure of their morphological irregularity and capable of distinguishing relatively transparent numerals (e.g., English twenty-one which has a clear relationship to the numerals twenty and one) from less transparent ones (e.g., Hindi ɪkkis ‘21’ which has a less clear relationship to the numerals ek ‘1’ and bis ‘20’). These metrics show the following generalizations:

  • Numeral systems of South Asia (specifically of Indo-Aryan languages) show considerably higher degrees of complexity than those of other languages of the world.
  • Higher complexity in numeral systems was preserved in the core Indo-Aryan linguistic region. Languages spoken at higher altitudes developed less complex systems, but this was largely an artifact of the adoption of vigesimal systems.
  • As with other languages’ systems, Indo-Aryan numeral systems display a negative relationship between cardinality and complexity/irregularity, but their overall complexity is higher and the dropoff in complexity as cardinality increases is less steep.

Data

Data for cross-linguistic numeral systems were taken from two sources:

  • UniNum (Ritchie et al. 2019), a collection of numerals between 0 and 100000000000, provided by Google and language experts. This data set was curated for text-to-speech purposes, and contains data from 186 speech varieties. Representations of numeral wordforms are orthographic, not phonemic.
  • South Asian Numerals Database (SAND; Kumari 2023b, 2023a), a collection of numerals ranging from 1 to 10000000 from 131 speech varieties of South Asia, to which we added data from 10 additional Indo-Aryan languages. Representations of numeral wordforms in SAND are phonemic, transcribed in the International Phonetic Alphabet (IPA). For additional languages, the systematic phonemic (but not necessarily IPA) representations provided in sources were used.

We used only data for the numerals 1–99 from both databases.

Metrics

Minimum Description Length (MDL)

In line with the information theoretic principle of minimum description length (MDL; Rissanen 1983), we seek the shortest set of combinable elements needed to generate all 99 numeral words of interest. We use a simplified version of models employed for morpheme and word segmentation (Goldsmith 2001; Creutz and Lagus 2007; Goldwater, Griffiths, and Johnson 2009) using expectation maximization (Dempster, Laird, and Rubin 1977) in order to segment each numeral form in each language into recurrent subword units such that the set of segmented units is minimized.

For each language, we randomly initialize the segmentation of each numeral word form \(w\). We do not allow numerals 1–9 to be segmented (as they tend to be simplex forms except in the case of systems with bases lower than 10, which are underrepresented in the data sample); we allow 11–19 and multiples of 10 to either be unsegmented (as in less transparent, more fusional forms like En. twelve, Sp. cuarenta ‘40’) or have a single segmentation index \(i \in \{2,...,|w|-1\}\), where \(|w|\) represents word form length, and \(i\) the index marking the start of the second subword unit (as in more transparent forms like Gm. Acht-zehn ‘18’, Jp. ni-jū ‘20’). All other numerals are forced to have a single segmentation index (as above) or two segmentation indices \((i,j) \in \{2,...,|w|-1\}:j \neq i\) Segmented units are placed in a cache \(\mathbb{W}\).

For each EM iteration \(t\), we randomly consider each word \(w\) representing the numerals 10–99, removing the currently segmented units \(\sigma(w)_z^{(t-1)}\) from the cache \(\mathbb{W}\). We then choose a new segmentation \(z^{(t)}\) (either no segmentation, one index, or two, depending on the conditions outlined above) such that \(\text{arg min}_{z^{(t)}} \mathbb{W} \cup \sigma(w)_{z^{(t)}}\). We stop when the description length reaches a minimum, or after 1000 iterations. Because this version of the EM algorithm converges on local and not global optima, we run this procedure 10 times per language, storing the shortest description length across runs. We note that this procedure does not account for allophonic processes, and may infer different subword elements that are underlyingly the same according to standard phonological analyses due to allophony, or orthographic variation.

This procedure yields a single MDL value for each language, representing the complexity of the language’s numeral system as a whole.

N-gram surprisal in context

We additionally operationalize the complexity of numeral systems using n-gram (specifically segmental/grapheme trigram) continuation surprisal, representing the unpredictability of phoneme or grapheme in context, i.e., given the two previous phonemes/graphemes (Piantadosi, Tily, and Gibson 2012; Dautriche et al. 2017), averaged within and across words. Numeral systems containing more recurrent, predictable elements are expected to exhibit lower surprisal. We compute the mean n-gram surprisal of individual numeral words conditioned on all other words in the system (training vs. held-out).

For each language, we compute the held-out trigram surprisal of each word form as follows. First, we prepend two instances of the beginning of string sequence and postpend one instance of the end of string sequence to each word form. We tabulate frequencies \(c()\) of all trigram and bigram segment/grapheme sequences in the training data. For the word form under consideration, we compute the continuation surprisal of each segment/grapheme at index \(i \in \{3,...,|w|\}\), \(s(w_i|w_{i-2...i-1}) = -\log \frac{c(w_{i-2...i}) + \alpha}{c(w_{i-2...i-1}) + V\alpha}\) (where \(\alpha=.1\) is an additive smoothing constant and \(V\) is the number of segment types), and average these values.

This procedure yields a surprisal value for each numeral in the system, conditioned on the other members of the system. These values can be averaged to represent the overall surprisal of the system at the language level.

Linear discriminative learning: production

We use a simplified version of the linear mapping approach (proposed in Baayen, Chuang, and Blevins 2018 et seq) to model the production of numeral forms (e.g. twelve) given an underlying semantic representation ({TENS=1,DIGITS=2}). Unlike the original version of this model, numeral forms’ meanings are represented by two one-hot vectors (comprising the tens- and digits-place value of a numeral) rather than continuous semantic vectors, which are concatenated together and make up rows of the meaning matrix \(\boldsymbol{S}\). Numeral forms are represented by a vector containing the counts of the trigrams they contain, which make up rows of the word form matrix \(\boldsymbol{W}\).

Form generation given a semantic representation \(\boldsymbol{s}_i\) proceeds as follows. We compute the least-squares solutions \(\boldsymbol{\hat{\beta}_{sw}}\) and \(\boldsymbol{\hat{\beta}_{ws}}\) that solves the equations \(\boldsymbol{S}_{-i}\boldsymbol{\hat{\beta}_{sw}} = \boldsymbol{W}_{-i}\) and \(\boldsymbol{W}_{-i}\boldsymbol{\hat{\beta}_{ws}} = \boldsymbol{S}_{-i}\), respectively.

We then compute a vector of weights \(\boldsymbol{s}_i^{\top} \boldsymbol{\hat{\beta}_{sw}}\), representing the association strengths of different trigrams (in reality their predicted counts, but in real-valued space) with the semantic representation \(\boldsymbol{s}_i\). We decode the form via beam search: starting with trigrams beginning with the start-of-sequence token, we consider the two trigrams forming valid continuations of the sequence with highest association strength, stopping when the end-of-sequence token is reached. For each candidate form \(\boldsymbol{\hat{w}}\), we compute the predicted meaning \(\boldsymbol{\hat{w}}^{\top} \boldsymbol{\hat{\beta}_{ws}}\), choosing the candidate form that shows highest correlation (Pearson’s \(r\)) with \(\boldsymbol{s}_i\). We measure the error rate between the predicted form and the true form using the Levenshtein distance between the two forms divided by the length of the longer form.

Linear discriminative learning: comprehension

To model the comprehension or discrimination of the meaning of a given form, we train two multinomial logistic classifers on the word form matrix \(\boldsymbol{W}_{-i}\) in order to predict the tens and digits label of \(\boldsymbol{w}_i\) with maximum probability according to the classifiers. We treat classification accuracy as a binary variable valued \(1\) if the tens and digits label are correctly predicted and \(0\) otherwise.

Results

Complexity metrics in a global perspective

Here, visualizations for different complexity metrics based on the UniNum dataset are given. Chinese varieties excluded from the surprisal map due to inflated surprisal values resulting from their ideographic writing system.

Below, first principal component resulting from PCA performed on these variables is displayed on a map:

PC1 explains 0.999 of the variance in the data, and shows a strong correlation with MDL.

Predicted complexity values for South Asia vs. rest of world:

Complexity metrics within South Asia

Here, visualizations for different complexity metrics based on the augmented SAND dataset are given.

Below, the first principal component resulting from PCA performed on these variables is displayed on a map:

PC1 explains 1 of the variance in the data, and shows a strong correlation with MDL.

Below, the first principal component is shown for all Indo-Aryan languages, with language labels (n.b. that some choices of coordinates, e.g., for Sanskrit, may not provide a good indication of the region in which languages are/were spoken):

Predicted complexity values for Indo-Aryan vs. non-Indo-Aryan:

Here, we list complexity values for individual Indo-Aryan languages, ordered according to PC1:

Name PC1 MDL surprisal PER accuracy
Hajong -21.7113033 30 0.9646738 0.2748347 0.8888889
Kudmali -13.7158920 38 0.9265155 0.1034741 0.8989899
Indus Kohistani -12.7137022 39 0.9252964 0.1777522 0.6464646
Shina -10.7142756 41 0.9157767 0.2882573 0.7171717
Rohingya -10.7097736 41 1.1734988 0.2519170 0.8383838
Palula -5.7208720 46 0.7473512 0.2228144 0.8282828
Torwali -4.7189559 47 0.8675831 0.1828304 0.8484848
Sinhala -3.7159348 48 1.0228114 0.2102460 0.8888889
Sanskrit -3.7058906 48 1.3595579 0.3908640 0.7272727
Phalura -1.7236683 50 0.7151789 0.1462739 0.8686869
Lambadi -1.7119757 50 1.2089042 0.1939510 0.8181818
Domaaki -0.7166503 51 0.7925570 0.2084482 0.2121212
Kashmiri -0.6998324 51 1.5990529 0.4662076 0.5858586
Halbi 0.2791241 52 0.8223804 0.1777302 0.7676768
Brokskad 2.2801786 54 0.8609005 0.2672853 0.7474747
Kalasha 5.2773222 57 0.8291410 0.1721655 0.8181818
Saraiki 9.2981383 61 1.6220775 0.5230163 0.5757576
Gaddi 9.3044439 61 1.9018381 0.5149403 0.5555556
Pahari 10.3034984 62 1.8754042 0.5291076 0.5757576
Sindhi 12.2987308 64 1.7280754 0.4603491 0.6262626
Sylheti 13.2981560 65 1.7159670 0.4379585 0.6060606
Dogri 13.3085068 65 2.0900982 0.5133833 0.4141414
Dhivehi 15.2871934 67 1.2829792 0.3996908 0.6666667
Konkani 15.3007209 67 1.8054765 0.4483577 0.4646465
Nagpuri 19.3004819 71 1.8868177 0.4430827 0.5656566
Kulvi 19.3062491 71 2.0972168 0.4965492 0.4747475
Nepali 20.2977425 72 1.7958055 0.4950057 0.6666667
Pakistani Punjabi 20.3025167 72 1.9261491 0.4812200 0.4141414
Urdu 21.3012741 73 1.8995732 0.4986164 0.4747475
Garhwali 22.2988057 74 1.8557361 0.4432907 0.5757576
Indian Punjabi 22.3021722 74 1.9742159 0.4576899 0.4949495
Kumaoni 23.2985458 75 1.8570291 0.4452969 0.5757576
Oriya 23.2997747 75 1.9242077 0.4530390 0.6161616
Panchparganiya 24.2970098 76 1.7924628 0.5057436 0.6060606
Gujarati 26.3018633 78 1.9887736 0.5379418 0.4949495
Khortha 27.2989620 79 1.9119964 0.4689459 0.5454545
Bengali 28.2975929 80 1.8402168 0.5197726 0.5252525
Assamese 29.2925140 81 1.6544109 0.4402032 0.5353535
Wagdi 29.3031549 81 2.0974679 0.5198141 0.5050505
Marathi 30.2974201 82 1.8531737 0.5289624 0.5151515
Marwari 32.3032315 84 2.1431204 0.5380146 0.5252525
Rajbongshi 33.2973761 85 1.8678471 0.5706885 0.4848485
Bagri 34.2946482 86 1.8224834 0.4731891 0.5757576

Relationship between complexity and other predictors

Relationships between complexity, elevation, and vigesimality are visualized below:

Predicted vigesimality values from a fitted model (gam(PC1 ~ vigesimal + s(elevation), data=merged[merged$Family=='Indo-Aryan',])):

Predicted slope of smooth function representing effect of elevation on PC1 from same model:

Complexity metrics within languages

UniNum, linear discriminative learning (production accuracy) by cardinality

(selected languages)

GAM smooths

Marginal slopes

Marginal effects of group

UniNum, linear discriminative comprehension accuracy by cardinality

(selected languages)

GAM smooths

Marginal slopes

Marginal effects of group

UniNum, surprisal by cardinality

(selected languages)

GAM smooths

Marginal slopes

Marginal effects of group

Sensitivity analysis on smoothing constant values

Correlations between surprisal values computed from the UniNum dataset under different smoothing constants are shown below:

Correlations between surprisal values computed from the SAND dataset under different smoothing constants are shown below:

References

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———. 2023b. “CLDF Dataset Derived from Mamta’s "South Asian Numerals Database" from 2023.” Zenodo. https://doi.org/10.5281/zenodo.10033151.

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